Transducer array with constant pressure, plane wave near-field



w. J. TRo'r-r 3,364,461

TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELD Jan. 16,1968 '7 Sheets-Sheet l Filed July 30, 1965 q" El Jan. 16, 1968 Wl J,TROTT 3,364,461

TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEARFIELD Filed July30, 1965 '7 Sheets-Sheet 2 fa ea da ca ba au fb eb db cb bb ab fc ec dccc bc ac fd ed dd cd bd ad fe ee de ce be ne ff ef df cf bf af INVENTORw//vF/ELD J. TROTT ATTORNEY Jan. 16, 1968 W. J. TROTT Filed July 30,1965 7 Sheets-Sheet 5 sI-IADING ELEMENT Isa a sHADED LINE ARRAY 52\SHADING ELEMENT I5b b SHADED LINE ARRAY 53N SRADING ELEMENT Isc c sHADEDLINE ARRAY sHAoING ELEMENT Isd d SHADED LINE ARRAY I 55X I SHADINGELEMENT Ise e sHADI-:o LINE ARRAY 56`\ sHADING ELEMENT Isf f SHADED LINEARRAY j 57\ sHAoING ELEMENT |59 sHADED LINE ARRAY i: 585 I sRAoINGELEMENT` Ish e sHADED LINE ARRAY 59\ I SHADING ELEMENT I5) SHADED LINEARRAY GO sHAoING r ELEMENT I5k c SHADED LINE ARRAY 6l\ a sHADINs ELEMENTIsm b SHADED LINE ARRAY 62\ sHAnING ELEMENT Isn c I sRADED LINE ARRAY G6SIGNAL souRcE 67 INVENTOR W/NF/ELD J. TROTT ATTORNEY W. J. TROTT Jan.16, 1968 TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELDFiled July zo, 1965 mm .w-m mmm '7 Sheets-Sheet 4 F m, l

INVENT OR W//VF/ELD J. TROTT ATTORNEY W. J. TROTT Jan. 16, 1968TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEARFIELD 7Sheets-Sheet 5 Filed July 30, 1965 ATTORNEY Jan. 16, 1968 w. J. TRoT'r3,364,461

TRANSDUCER ARRAY WITH CONSTANT PRESSURE, PLANE WAVE NEAR-FIELD FiledJuly 50, 1965 '7 Sheets-Sheet 6 2O log P O 0.5 I.O

W/NFIELD J. TROTT Y/ff/wiwg,

ATTORNEY W. J. TROTT Jan. 16, 1968 TRANSDUCER ARRAY WITH CONSTANTPRESSURE, PLANE WAVE NEAR-FIELD '7 Sheets-Sheet '7 Filed July 30, 1965ATTORNEY United States Patent Oiice 3,354,461 Patented Jan. 16, 19683,364,461 TRANSDUCER ARRAY WITH CONSTANT PRES- SURE, PLANE WAVENEAR-FIELD Winfield J. Trott, Orlando, Fla., assignor to the UnitedStates of America as represented by the Secretary of the Navy Filed July30, 1965, Ser. N0. 476,214 28 Claims. (Cl. 340-6) ABSTRACT F THEDISCLOSURE A shaded transducer array wherein the individual elements ofthe array are shaded to produce a constant, plane wave near-fieldextending over the aperture of said array. The shading is such that thesensitivities of the elements increase from the extremities toward thecenter of the array according to the coefficients of a summed binomialprobability distribution function.

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

This invention relates to a shaded transducer array and moreparticularly to a transducer array that lies in the Y-Z plane of arectangular coordinate system and radiates a wave in the direction ofthe X axis wherein the lndividual elements of the array are shaded soasto produce a plane wave with constant Y and Z components in the nearfield of the array.

Where the elements of the array are electroacoustic transducers, thearray produces a constant pressure, plane wave near-field. Where theelements are antennae, the array produces a constant electric field,plane wave near-field.

The present invention finds particular utility in the calibration ofelectroacoustic transducers from data obtained by measurements made inthe near field of the unknown transducer and Will be described withparticular reference to such use. However its utility is not limitedthereto. Rather, the shading taught by the present invention can be usedanytime that it is desired to obtain an electroacoustic array having aconstant pressure, plane Wave nearfield or an electromagnetic arrayhaving a constant electric field, pl-ane wave near-field.

In the calibration of electroacoustic transducers, particularly thosefor use underwater, much attention has recently been focused onobtaining the calibration data from measurements made in the near fieldof the unknown transducer so that the inadequate dimensions and nonidealboundaries of existing calibration facilities do not prohibit the usethereof for the calibration of large, low frequency transducers whichare becoming increasingly important for both civilian and militarypurposes. To date, the methods used to obtain the calibration data frommeasurements made in the near field have required that numerous delicatemeasurements be made thus requiring that considerable skill be employedand allowing significant room for error.

A general purpose of this invention is to provide an electroacoustictransducer array which can be used in the calibration of electroacoustictransducers in such a manner that all of the known ladvantages ofCalibrating electroacoustic transducers from measurements made in thenear field are retained while the aforedescribed disadvantages of soCalibrating transducers are eliminated. To attain this, the presentinvention contemplates shading the individual elements of anelectroacoustic transducer array so that the array has a constantpressure, plane wave near-field and making the measurements necessary tocalibrate the unknown electroacoustic transducer while it is maintainedwithin this constant pressure, plane wave near-field.

An object of the present invention is to provide an array having aunique near-field.

Another object is to provide a transducer array which lies in the Y-Zplane of a rectangular coordinate system and which produces a plane waveradiating in the X-direction, which wave has constant Y and Z componentsin the near field.

A further object of the invention is the provision of a shadedelectroacoustic transducer array which produces a constant pressure,plane wave near-field.

Still another object is to provide a shaded antenna array which producesa constant electric-field, plane Wave nearfield.

A still further object is to provide a simple and accurate method ofobtaining the data necessary to calibrate an electroacoustic transducerfrom measurements made in the near field of the transducer.

Other objects and many of the attendant advantages of this inventionwill be readily appreciated as the same becomes better understood byreference to the following detailed description when considered inconnection with the accompanying drawings in which like referencenumerals designate like parts throughout the figures thereof andwherein:

FIGS. 1A and 1B, when taken together, illustrate an electroacoustic linetransducer shaded according to the present invention;

FIG. 2 illustrates the shading of one quadrant of a plane array havingapproximate circular symmetry;

FIG. 3 represents a plane array having approximate circular symmetry andbeing shaded according to the pres ent invention;

FIG. 4 illustrates the shading of one quadrant of a plane array havingsquare symmetry;

FIGS. 5a and 5b, when viewed together, show one quadrant of a planearray having square symmetry;

FIG. 6 is an optimum plot of the relative sound pressure at the centerof an array having approximate circular symmetry vs. frequency with theplot normalized for any size array;

FIG. 7 illustrates the method of using the array of the presentinvention to calibrate an unknown transducer; and

FIG. 8 shows an antenna line array shaded according to the instantinvention.

Turning now t0 FIGS. 1A and 1B which, when viewed together, show atwelve element basic electroacoustic line array l5 shaded according tothe present invention. Shading elements 19-30 are respectively coupledin series with parallel energized electroacoustic transducers 34-45. Itshould be understood that transducers 34-45 can be piezoelectric,magnetostrictive, or variable reluctance types and that shading elements19-30 can be any type of shading means known in the art to beappropriate to shade the particular type transducer used. For instance,if transducers 34-45 are piezoelectric, shading can be obtained byplacing appropriate size capacitors in series with each of thetransducers or by etching away part of one or more of the electrodesmaking electrical contact with the respective transducer. If thetransducers are magnetostrictive, shading can be obtained by properlyselecting the number of turns in the winding on each ofthe cores or byvarying the amount of magnetostrictive material in eachV of the cores.These examples are not exhaustive but, rather, illustrate that theshading may be obtained by any means appropriate to control thesensitivity of the individual transducers.

lt should be understood that the line array of FIG. 1 could be used as areceiver as well as a transmitter. When operating as a receiver, theoutput voltage would appear across a load (not shown) connected betweenterminals 46 and 47.

It is known that if shading in the form of a Gaussian function is usedto shade an array, the pressure along the beam axis of the array will beconstant in the near iield and all minor lobes in the far elddirectivity of the array will be suppressed. But, shading of this formdoes not provide the constant pressure, plane wave near-eld desiredbecause the pressure falls off rapidly for positions in the aperture ofthe array that are 01T the beam axis.

Rather, the instant invention contemplates the use of a shading functionthat will cause the first minor lobe in the far eld directivity to beattenuated only slightly and all successive minor lobes to be suppressedto a much greater extent. An array so shaded will have a pressuremaximum on the beam axis at the far extreme of the near field and amaximum at the arrays periphery. Therefore, the pressure will be keptconstant over a greater portion of the aperture than when all minorlobes are suppressed as in shading in the form of a Gaussian function.

It has been found that if a line array is symmetrically shaded about thecenter with the sensitivity of the individual transducers thereofdecreasing from the center toward the respective ends thereof accordingto the values of a binomial probability distribution function, thedesired slight attenuation in the rst minor lobe and much greatersuppression of all subsequent minor lobes in the far field directivitywill be obtained. The values of the binomial probability distributioncan be determined from the formula:

where S=the transducer sensitivity;

This formula will be recognized as the formula for determining theprobability of at least r occurrences in n independent binomial trials,when the probability in any single trial is 0.5.

When the line array of FIG. l is shaded in this manner, the shadingelements 19-30 are chosen so that transducers 34--45 have sensitivitiesof 0.03, 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19, and0.03 respectively. These cO- efficients are obtained by setting n inEquation l equal to 5. Of course, n would not have to be equal to 5 butcould be any value appropriate to the number of transducers in thearray. One of many published tables in which these coefficients areavailable is; National Bureau of Standards, Applied Mathematics SeriesNo. 6, Tables of the Binomial Probability Distribution, U.S. GovernmentPrinting Oice, Washington, DC., 1950. This table gives coeflicients forvalues of n from 2-49.

It should be understood that there is quite a bit of latitude availablein adding unshaded transducers (that is, transducers unacected by anyshading so that their normalized sensitivity coeicients are equal to1.0) in the center of the array without destroying the constantpressure, plane wave near-field. For example, if a fourteen element linearray were desired, the individual transducers thereof could havesensitivities from the left to right ends thereof of 0.03, 0.19, 0.5,0.81, 0.97, 1.0, 1.0, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19 and 0.03respectively.

It is also possible to derive the coeicients of the shading function ofthe present invention by taking the coeiiicients of a binomial serieshaving a power equal to n and replicating the coeicients of the series mtimes where m=n+c and c equals the number of unshaded transducers in thecenter of the array. For example, the coelicients of the shadingfunction for the line array of FIGS. lA and 1B are from left to right,0.03, 0.19, 0.5,

d 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19 and 0.03. Thesecoetiicients represent the normalized result of taking the coefficientsof a binomial series having a power rt=5 and replicating them m=11lc=7times.

The general equation for the directivity of a line array shadedaccording to the present invention is:

sin me D um singb s (2) where P: pressure;

di= sin 0,

d=element spacing; t=the wavelength at which the line is operating;

and

zthe angle in the plane containing the line and the beam axis of theline between the plane normal to the line which also contains the beamaxis and the plane that contains a distant measuring point andintersects the line at the beam axis.

Applying Equation 2 to the line array shown in FIG.

l, it is seen that the directivity thereof is:

sin 7b p: 7 sintb COSS qs n to have the plane array have the samefar-eld directivity as that of the line array.

Consider a plane array lying in the Y-Z plane of a rectangularcoordinate system with each row of individual transducers thereof beingparallel to the Y axis and containing a given number of transducersspaced from one another a distance d1 and with a given number of columnsof individual transducers spaced from one another a distance d2 and eachcolumn being parallel to the Z axis. Further, consider a line arraysuperimposed over each row and column of the plane array with the linearrays superimposed over the rows containing the same number oftransducers as contained in the rows with the transducers in the linearrays spaced from one another the same distance d1 as the transducer inthe rows and with the line arrays superimposed over the columnscontaining the same number of transducers as contained in the columnswith the transducers in the line arrays spaced from one another the samedistance d2 as the transducers in the columns. Then, it is known that ifthe transducers of the plane array are shaded so that their respectivesensitivities are equal to the product of the sensitivities of the twotransducers superimposed thereover, the plane array will haveapproximate circular symmetry and a far-field directivity the same asthat of the shaded line arrays.

Thus, if the shading function for the line array of FIGS. 1A and 1B isrepresented, from left to right, by

a, b, c, d, e, f, f, e, d, c, b, a, the individual transducers for atwelve by twelve plane array having approximate circular symmetry andthe same directivity as that of the line array would have sensitivitiesas shown for the upper right quadrant of such as array in FIG. 2. As isobvious from the above, if the identically shaded line arrays used toderive the shading function for the plane array have constant pressure,plane wave near-fields the plane array will also have a constantpressure, plane wave near-held.

FIG. 3 represents a typical embodiment of a twelve by twelve plane arrayhaving approximate circular symmetry of shading elements and beingshaded according to the.

present invention. Line arrays a-15h, 15j, 15k, 15m; and 15m are allshaded identically to each other and to line array 15 of FIGS. 1A and1B. For convenience of expression, the 0.03, 0.19, 0.5, 0.81, 0.97, 1.0,1.0, 0.97, 0.81, 0.5 0.19, 0.03 shading coecients of line array 15 ofFIGS. 1A and 1B and 15a-15h, 15j, 15k, 15m, and 15n and of FIG. 3 arerespectively represented by a, b, c, d, e, f, f, e,d,c,b,a.

Shading elements 51-62 are connected in series with the inputs to linearrays 15a-15h, 15j, 15k, 15m, and 1511 respectively and have shadingcoefficients of a, b, c, d, e, f, j", e, d, c, b, a respectively. Thus,the effective shading of each of the transducers will be the product ofthe shading coeflicient of the shading element connected in series withthe input to the line array containing the transducer and of the shadingcoefficient of the shading element within the line array that isconnected in series with the input to the transducer. For example, theeffective shading of the individual transducers within line array 15awill be, from left to right, aa, ab, ac, ad, ae, af, af, ae, ad, ac, ab,aa.

When the plane array of FIG. 3 is operating in a transmitting mode,signal source 65 will impress an input signal across terminals 66 and67. When the array is operating in a receiving mode, the output voltagewill appear across a load (not shown) connected between terminals 66 and67.

It should be clear that FIG. 3 does not :represent the only possibleembodiment of a plane array having approximate circular symmetry.Rather, such an array could be composed of any type of electroacoustictransducer `and by Shading appropriate to the transducer used. However,it will be noted that the sensitivities of the individual elements inthe plane array all satisfy the equation:

S'---Slsa (3) where S=The sensitivity of a transducer in the planearray;

S1=the sensitivity of the transducer in a first line array whichoccupies the same position with respect to the other transducers in saidfirst line array as said transducer of the plane array occupies withrespect to the other transducers of the row of said plane array in whichit is included, said first line array containing the same number oftransducers as contained within said row being spaced identically tosaid transducers in said row, lthe sensitivities of the transducers insaid first line array being shaded so as to increase from theextremities toward the center thereof according to the coefficients of abinomial probability distribution function; and

S2=the sensitivity of the 4transducer in a second line array whichoccupies the same position with respect to the other transducers in saidsecond line array as said transducer of said plane array occupies withrespect to the other transducers of the column of the plane array inwhich it is included, said second line array containing the same numberof transducers as contained within said column being spaced identicallyto the transducers in said column, the sensitivities of the transducersin said second line array being shaded so as to increase from theextremities toward the center thereof according to the coefficients of abinomial probability distribution function.

FIG. 4 represents the sensitivities of the transducers of a plane arrayhaving square symmetry wherein the plane array is shaded according tothe present invention. Only the upper-right quadrant is shown. However,it can be seen from FIG. 4 that the full array has families ofrectangular, or more particularly square, groups of transducers havingthe same sensitivities with the groups so arranged that each transducerlies in a horizontal row and a vertical column of transducers.Furthermore, it can be seen from FIG. 4 that the groups are so arrangedthat the perpendicular bisectors of the sides of the rectangles that aredefined by the transducers having the same sensitivity all meet at acommon point.

In order that Equation 2 represent the far-field directransducers in theplane array satisfies the equation:

1 th 1 S s. s.

Jp 7LiL ,L X=jLI1 lolx-l) x where jp identifies any one of the groups bythe number of groups said one of said groups is removed from said commonpoint and :1, 2, 3, tp;

tpzthe total number of groups;

Sjp=the sensitivity of the transducers in the jp group;

jL identifies any one of the transducers in said line array by thenumber of transducers said one of said transducers is removed from thecenter of said line array and :jpg

njL=the number of transducers the jL transducer is removed from thecenter of said line array;

SjL=the sensitivity of the jL transducer;

tL=onehalf the total number of transducers in said line array;

X identifies any one of the transducers in said line array;

nx=the number of transducers the x Itransducer is removed from thecenter of said line array; and

SX=the sensitivity of the x transducer.

FIGS. Sa and 5b, when viewed together, show the right quadrant of a l2by 12 plane array having square symmetry and being shaded according tothe present invention. As shown, the shading is achieved by placingappropriate sized shading elements in series with each of thetransducers. Such an arrangement would obtain where the shading elementsare capacitors and the transducers are piezoelectric elements. However,it should be understood that the present invention is not limited to theuse of piezoelectric transducers or series capacitor shading but,rather, may be used in conjunction with any type transducer andappropriate means to control the sensitivity of the transdncer used.

As shown in FIGS. 5a and 5b, the array is connected so as to operate ina transmitting mode with signal source 75 impressing an input voltageacross terminals 76 and '77. However, the array could also be used in areceiving mode in which case signal source 75 would be removed and theoutput voltage would appear across a load (not shown) connected betweenterminals 76 and 77.

There are a set of conditions pertaining to the upper and loweroperating frequency limitations that the line array, plane array havingcircular symmetry, and plane array having square symmetry all mustsatisfy if they are to produce a constant pressure, plane wavenearfield. To express this set of conditions, it is necessary to definea radius R as being 1/2 the distance between the points each side ofcenter of the line array around which a symmetry exists with respect tothe sensitivities of the transducers on the respective sides of thecenter Whose sensitivities are affected by the shading. For example, theshading of the basic shaded array of FIGS. 1A and 1B is, from left toright, 0.03, 0.19, 0.5, 0.81, 0.97, 1.0, 1.0, 0.97, 0.81, 0.5, 0.19,0.03. It can be seen that the shading does not affect the transducershaving sensitivities of 1.0 and that there is a symmetry of shadedelements on the respective sides of the center around the elementsshaded to 0.5. Thus, if each element is considered to be spaced from theadjacent element a distance d, 1/2 the distance between the shadedelements around which symmetry of shaded elements exists would be equalto 1/2 the distance between the elements shaded to 0.5 which would beequal to 75l/2.

Then, the upper and lower operating frequency conditions can beexpressed as:

where d=spacing between individual transducers in the line array;

)t the wavelength of the operating frequency; and

R=1/2 the distance between the points each side of the center of theline array around which a symmetry exists with respect to thesensitivities of the transducers on the respective sides of said centerwhose sensitivities are affected by said shading.

A computation of the relative pressure vs. frequency at the center of anarray having circular symmetry shows the greatest pressure amplitudevariation that will exist whether the array has approximate circular orsquare symmetry provided that both arrays are derived from the samebasic shaded line.

An array having approximate circular symmetry can be considered to be apiston of unity source strength density with a radius equal to R (whichwas defined above to be 1/2 the distance between the shaded elements ofthe basic line array around which symmetry of shaded elements exists)with shading superimposed on this piston by the addition of ringsources.

The relative near-held pressure at the center of an unshaded piston ofunity source strength density is given p=2sin1/2kRexpi(wt-l1/z1r1/2 R)(6) where C=21r/)\; \=the wavelength of the operating frequency of thepiston;

R=the radius of the piston; and w=the operating frequency of the pistonin rad/sec.

The relative near-field pressure at the center of a ring source is givenby:

p=2wi sin l/zkd exp [(wt-l-l/zn-*kRQ (7) where W=the source strengthdensity of the ring; k=21r/;

From Equations 6 and 7 it is possible to compute the relative pressureat the center of a plane array having approximate circular symmetry. Forexample, the plane array of FIG. 3, which was derived from the basicshaded line array of FIGS. 1A and 1B, can be represented by a pistonhaving unity source strength density and a radius equal to the distancebetween the center Of the line array and the element of 0.5 sourcestrength density with tive ring sources, each of a width d equal to thedistance between adjacent elements in the basic line array, superimposedthereon. The rst ring has a source strength density of 0.03 and anaverage radius Ri equal to the distance from the center of the linearray to the element of source strength density equal to 0.97 in theline array; the second has a source strength density of 0.19 and anaverage radius Ri equal to the distance from the center of the linearray to the element of source strength density equal to 0.81; the thirdhas a source strength density of 0.5 and an average radius R1 equal tothe distance from the center of the line array to the element of sourcestrength density equal to 0.5; the fourth has a source strength of 0.19and an average radius Ri equal to the distance from the center of theline array to the element of source strength density equal to 0.19; andthe fifth has a source strength density of 0.03 and an average radius Riequal to the distance from the center of the line array to the elementof source strength density equal to 0.03.

When ring sources of source strength densities of 1/2, iWl, iWg, iWa,etc., are superimposed on a piston with a radius R and unity sourcestrength density, the pressure at the center for half of the ringsources, n odd, including the ring with a source strength density of 1/2and radius R, is given by:

The remaining half of the ring sources n even, will include no ring ofsource strength density 1/2 and radius R. For this remaining half, R isdefined as the distance from the common center of the rings to a pointhalf way between the ring with a source strength density greater than1/2 and the ring with a source strength density less than 1/2 and therelative pressure at the center of these rings is given by:

It will be observed that Equations 8 and 9 are based on a steppeddensity distribution rather than a point density distribution. However,this distribution is valid for a square array having approximatecircular symmetry.

Using Equations 8 or 9 it is possible to compute the relative soundpressure vs. frequency at the center of an array having approximatecircular symmetry. In the example involving the array having approximatecircular symmetry of FIGS. 2 and 3, which was derived from the basicshaded line array of FIGS. 1A and 1B, w1 and wz of Equations 8 and 9 areequal to 0.19 and 0.03 respectively.

FIG. 6 is an optimum plot of Equations 8 and 9. The abscissa isnormalized in terms of R so that the plot applies to any size arrayhaving approximate circular symmetry which is shaded according to thepresent invention. FIG. 6 can be used to determine the optimum shadingfunction for any array Whether the array be a line array or a planearray having approximate circular or square symmetry. This can be doneby shading a plane array having approximate circular symmetry accordingto a binominal probability distribution function and then adding ortaking away unshaded elements from the center of the array until a plotof Equations 8 or 9 approximately yields the curve of FIG. 6. The linearray can then be obtained from the plane array having approximatecircular symmetry through the application of Equation 3 and the planearray having square symmetry can be derived from the line array throughthe application of Equation 4.

Measurements show that the diameter of the aperture for a constantpressure, plane wave near-held is approximately the distance between,the transducers having normalized sensitivities of approximately 0.8 andthat the axial depth of `the constant pressure, plane wave near field isapproximately RZ/, where R equals the distance from the center of thebasic line array to the shaded element one side of said center aroundwhich symmetry of shaded elements the same side of center exists and=the yvavelength of the operating frequency of the a array.

When it is desired to apply the present invention in the field ofelectroacoustics, the line array, plane array having circular' symmetry,and plane array having square symmetry can all he constructed in any o-fthe manners commonly used in the electroacoustic transducer art. Forexample, the individual elements may be mounted in oillled tubing ormolded into rubber or plastic jackets and mounted according toconventional practice. Often it is desirable to obtain electricalshielding. This can be done in any way Well known in the art such as bymounting the array between wire screens.

It should be understood that the showing of 12 by 12 plane arrays inFIGS. 3 and 4 is not meant to constitute a limitation upon the scope ofthis invention. There is no necessity that the number of transducers inthe rows equal the number of transducers in the columns for an arraycould be rectangular in shape with one set of binomial probabilitydistribution coet'icients along the rows and another set along thecolumns.

If an electroacoustic transducer array shaded according to the presentinvention is to be used as a measuring array to calibrate unknownelectroacoustic transducers, the array must be acoustically transparent,This is accomplished by maintaining the operating `frequency at leastone octave below the resonant frequency of the individual transducers ofthe array and by making each of these transducers small in comparison tothe Wavelength of the operating frequency.

Also, the array must be calibrated, meaning that the free-field voltagesensitivity M, the near-field transmitting current response Sp, and theratio of the free-field voltage sensitivity to the effective area of thearray M/A must be determined.

Within the range of frequencies, 4 kc.-l2 kc., in which the array of thepresent invention is of particular interest for calibration, thefree-held voltage sensitivity M is the same as the free-field voltagesensitivity of the individual elements of the array.

The near-field transmitting current response of the array, Sp, can bemeasured by placing a calibrated transducer in the near field of thearray and operating the transducer in its receiving mode and the arrayin its transmitting mode. Then:

Where A :the effective area vof the array, and pc=the characteristicimpedance of the environment.

Then,

Sp=M/Jp=M(pc/2A) (12) Ms can be expressed in terms of the far-fieldtransmitting current response, Ss, of the calibrated transducer and thespherical wave reciprocity parameter JS. By denition,

J,=2D t/pc (13) where D==the reference distance for the far-fieldtransmitting current response; )\=the wavelength of the operatingfrequency;

and

pc--the characteristic impedance of the environment.

Substituting Equations 12 and 14 into Equation l0 yields,

Espa 2SSDM (15) The ratio Es/I is the transfer impedance Ibetween thecalibrated transducer when operating as a receiver and the Mpc/2A) Cill@ array when operating as a transmitter. When the transducer isoperating as a transmitter and the array is operating as a receiver, thetransfer impedance is E/,Is, Where E equals the open circuit outputvoltage of the array and Is equals the current driving the calibratedtransducer.

Since the acoustical reciprocity theorem applies,

Substituting Equation 16 into Equation l5 and solving for the ratio M/Ayields:

Xios. E 17) The sound pressure represented by the product ISSS can bemeasured by means of a calibrated hydrophone placed in the far iield ofthe small calibrated transducer when operating in its transmitting modesince where Eh=the open circuit output voltage of the calibratedhydrophone, and

Mh=the free-eld sensitivity of the calibrated hydrophone.

In practice, only the measurements represented by Equations 17 and 18need, in fact, be made. These measurements provide all the informationnecessary to determine the near-field transmitting current response, Sp,and the ratio of the free-field voltage sensitivity M to the electivearea A.

When an array having a constant pressure, plane wave near-field iscalibrated, it can be used to calibrate any unknown transducer whosevolume is smaller than that `of the constant pressure, plane Wavenear-field of the array.

FIG. 7 illustrates a method of obtaining the data to calibrate anunknown transducer 81 through the use of a calibrated array 32incorporating the present invention so as to have a constant pressure,plane wave near field. The only limitations upon the use of array 82 tocalibrate an unknown transducer 81 are that the Volume of transducer 81be no greater than the volume 83 of the 4constant pressure, plane wavenear-field of array 82, that the unknown 81 be substantially within thisconstant pressure, plane wvave near-field, and that the operatingfrequency of unknown 81 be within the range of operating frequencies ofarray 82. Thus, it is clear that an array shaded according to theinstant invention may be designed to calibrate most any size transducer.

As an approximation, the volume represented .by 83 can be considered tohave a diameter equal to the distance between the transducers in thebasic shaded line array having sensitivities of approximately 0.8 and toextend to a distance from the face of array 82 of RZ/k where R equalsthe distance from the center of the basic line array to the shadedelement one side of center of the basic line array around which symmetryof shaded elements on that side of center exists and )t equals theWavelength of the operating frequency of the array.

The free-field voltage sensitivity MX of unknown transducer 81 is givenby MX=fEx/SPI (19) where Ex=the open circuit output voltage of theunknown;

SP1-the near-field transmitting current response of the a1'- ray; and

I=the current driving the array.

The measurements necessary to determine MX can be made in the embodimentshown in FIG. 7 by putting switch 84 in contact with terminal 85 so thatvoltmeter 86 will read EX and by putting switch 91 in contact withterminal 92 so that ammeter 93 will read the driving `current I suppliedby source 94, Sp is determined in the calibration of array 82.

1i il.

The far-iield transmitting current response of the unknown transducer l,SSX, can be mathematically derived from the near-field transmittingcurrent of the unknown SPX, the plane wave reciprocity parameter Jp, andthe spherical wave reciprocity parameter JS.

With the unknown Si operating as a so-urce and the calibrated array S2operating as a receiver, the near-field transmitting current response ofthe unknown is given by SML-PHx VJ here p=the average sound pressureproduced by the unknown which is measured by the array; and lxzthecurrent driving the unknown.

Since the average pressure p exists over an eective area AX which isless than the effective area A of the array, the pressure measured bythe array is P=(E/M) (f1/Ax) where Ezthe output voltage of the array;and M=the free-field voltage sensitivity of the array.

Therefore,

5px: (E/MIX)(A/AX) (20) The far-field transmitting current response Ssxof the unknown is related to the near-field transmitting currentresponse Spx by the ratio of the plane-wave reciprocity parameter 1p tothe spherical-wave reciprocity parameter J5 Thus,

SsX=EA/MIXD (21) where E :the open circuit output voltage of the array;

A :the effective area of the array;

M- ethe free-field voltage sensitivity of the array; Ix=the drivingcurrent of the unknown;

D=the reference distance (generally 1M); and Azthe wavelength of theoperating frequency.

The measurements necessary to determine Ssx can be made in theembodiment shown in FIG. 7 by putting switch 91 in contact with terminal96 so that voltmeter 97 will read E and by putting switch 84 in contactwith terminal 98 so that ammeter 99 will read the driving current ixsupplied by source 100. M and A are determined in the calibration of thearray and D is generally 1M.

The directivity of the unknown transducer 81 can also be determined oymeasurements made in the near field. Since the acoustical reciprocityprinciple applies, the directivity of the unknown can be measured byoperating it as a source and the array as a receiver and by making pointby point measurements of the sound pressure incident upon the calibratedarray as the unknown transducer is rotated in such a manner that itsbeam axis is rotated through 360 while being maintained in thehorizontal plane which contains the beam axis of the array.

Other acoustic parameters of the unknown 81, such as source level,transmitting voltage response, and transmitting power response, can alsobe determined from measurements made in the constant pressure, planewave near-field of the array 82.

FIG, 8 illustrates how the present invention may be applied to anantenna array. A line array comprising dipoles 111-122 is shown.However, it should be understood that the present invention could beapplied to a plane antenna array since the antenna array is analogous tothe transducer array.

Dipoles 1111-1 22 are respectively fed in parallel from source 137through shading elements 12S-136. Shading elements 12S-136 may be coilsor any of the other known means of controlling the sensitivity of adipole. As shown, the shading function for the 12 element antenna linearray is, from left to right, 0.03. 0.19, 0.5, 0.81, 0.97, 1.0, 1.0,0.97, 0.81, 0.5, 0.19, 0.03 which will be recognized as the same shadingfunction as that for the 12 element transducer line array shown in FIGS.1A and 1B. Similarly, a plane antenna array having approximate circularsymmetry could be constructed as shown in FIG. 3 by substituting dipolesfor the transducers shown and by using appropriate shading means and,also, a plane antenna array having square symmetry could be constructedas shown in FG. 5 by the substitution of dipoles and the use ofappropriate shading means.

lith the substitution of `dipoles for electroacoustic transducers and ofshading means appropriate to dipoles, an antenna array shaded accordingto the present invention is subject to the same design limitations, suchas requisite distance between elements and operating frequency, as atransducer array so shaded.

An antenna array shaded according to the present invention has aconstant electric field, plane wave near-field which is analogous to theconstant pressure, plane wave near-eld that an electroacoustictransducer array has when shaded according to the present invention.

Thus, the present invention provides an antenna array which has aconstant electric field plane wave near-field and an analogouselectroacoustic transducer array which has a constant pressure, planewave near-iield. Furthermore, the present invention provides a method ofcalibrating an electroacoustic transducer from measurements made in itsnear-field.

Obviously many modifications and variations of the present invention arepossible in the light ot the above teachings. It is therefore to beunderstood that, within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described.

What is claimed and desired to be secured by Letters Patent of theUnited States is:

1. An electroacoustic transducer array comprising:

a plurality of transducers;

means for shading the sensitivities of said transducers so that theirsensitivities increase from the extremities toward the center of saidarray according to the coeiicients of a summed binomial probabilitydistribution function having the general formula X (Dromwhere nzthenumber 0E independent binomial trials and r=1, 2, 3 n; so that saidarray has a constant pressure, plane wave near-field extending acrossthe aperture of Said array.

2. The electroacoustic transducer array of claim 1 wherein said array isa line array.

3. The electroacoustic transducer array of claim 1 wherein said array isa plane array.

4. An electroacoustic transducer line array comprising:

a plurality of electroacoustic transducers;

means for shading said transducers so that their sensitivities increasefrom the extremities toward the center of said line according to thecoefficients of a summed binomial probability distribution function sothat said line has a constant pressure, plane wave near-field extendingalong said line.

5. The line array of claim 4 wherein said transducers are electricallycoupled in parallel.

6. An electroacoustic transducer line array having a constant pressure,plane wave near-field and an operating frequency comprising:

a plurality of electroacoustic transducers equi-distantly spaced formingsaid line array;

means for shading said transducers so that their sensitivities increasefrom the extremities toward the center of said line array according tothe coe'icients of a binomial probability distribution function havingthe general formula (Drown n=the number of independent binomial trials,and r=1,2,3 ...n; said operating frequency being such that where whered=the distance between adjacent transducers in said line array, -:thewavelength of said operating frequency and R=the distance between thepoints each side of the center of said line array around which asymmetry exists with respect to the sensitivities of the transducers onthe respective sides of said center whose sensitivities are affected bysaid shading.

'7. The line array of claim 6 wherein said transducers are electricallycoupled in parallel.

8. The line array of claim 6 wherein:

each of said transducers has approximately the same resonant frequency;

said operating frequency is at least one octave below said resonantfrequency; and

the size of said transducers is small in comparison to said distancebetween adjacent transducers;

whereby said array is acoustically transparent.

9. An electroacoustc transducer plane array comprisa plurality ofhorizontal rows of transducers lying in a common plane;

each of said rows containing a plurality of transducers:

the transducers in each of said rows being vertically aligned with thetransducers in each of every other of said rows so that a plurality ofvertical columns of transducers are formed;

means for shading each of said transducers so that its sensitivitysatisfies the equation S=S1S2 where S--the sensitivity of the transducerin the plane array; S1=the sensitivity of the transducer in a first linearray which occupies the same position with respect to the othertransducers in said iirst line array as said transducer of said planearray occupies with respect to the other transducers of the row in whichit is included, said first line array containing the same number oftransducers as contained within said row being spaced identically tosaid transducers in said row, the sensitivities of the transducers insaid first line array being shaded so as to increase from theextremities toward the center thereof according to the coefficients of abinomial probability distribution function; and S2=the sensitivity ofthe transducer in a second line array which occupies the same positionwith respect to the other transducers in said second line array as saidtransducer of said plane array occupies with respect to the othertransducers of the column in which it is included, said second linearray containing the same i number of transducers as contained withinsaid column being spaced identically to the transducers in said column,the sensitivities of the transducers Ain said second line array beingshaded so as to increase from the extremities toward the center thereofaccording to the coefficients of a binomial probability distributionfunction.

10. The plane array of claim 8 wherein:

said plane array has the same number of rows as columns;

the transducers'in said first and second line arrays are shaded so thattheir sensitivities increase from the extremities toward the centers ofthe respective line arrays according to the coefficients of the samebinomial probability distribution function.

11. An electroacoustic transducer plane array having circular symmetry,a constant pressure, plane Wave neareld and an operating frequencycomprising:

a plurality of equidistantly spaced horizontal rows of transducers lyingin a common plane;

each of said rows containing the same number of trans` ducers and thetransducers within each row being spaced the same distance apart;

said rows being vertically aligned so that each transducer in a row isaligned in a vertical column with a transducer from each of every otherrow;

means for shading each of said transducers so that its sensitivitysatisfies the equation S=S1S2 where S=the sensitivity of the transducerin the plane array; S1=the sensitivity of the transducer in a first linearray which occupies the same position with respect to the othertransducers in said first line array as said transducer of said planearray occupies with respect to the other transducers of the row of saidplane array in which it is included, said first line array having thesame operating frequency as said plane array and containing the samenumber of transducers as contained within said row spaced the samedistance apart as the transducers in said row, the sensitivities of thetransducers in said iirst line array being shaded so as to increase fromthe extremities toward the center thereof according to the coeflicientsof a binomial probability distribution function having the generalformula (www where (wwwrun-ru n=the number of independent binomialtrials, and r=l, 2, 3 n; and S2=the sensitivity of the transducers in asecond line array which occupies the same position with respect to theother transducers in said second line array as said transducer of saidplane array occupies with respect to the other transducers of the columnof the plane array in which it is included said second line array havingthe same operating frequency as said plane array and containing the samenumber of transducers as contained within said column of said planearray spaced the same distance apart as the transducers in said column,the sensitivities of the transducers in said second line array beingshaded so as to increase from the extremities toward the center thereofaccording to the coeliicients of a binomial probability distributionfunction having said general formula: said operating frequency beingsuch that where d=the distance between adjacent transducers in one ofsaid line arrays, ?\=the wavelength of said operating frequency, andR=the distance between the points each side of the center of said one ofline arrays around which a symmetry exists wit-h respect to thesensitivities of the transducers on the respective side of said centerwhose sensitivities are affected by said shading.

12. The plane array of claim 11 wherein:

each of said transducers of said plane array has approximately the sameresonant frequency;

said operating frequency is at least one octave below said resonantfrequency;

the size of said transducers of said plane array is small in comparisonto said distance between adjacent transducers of said plane array; and

whereby said plane array is acoustically transparent.

i 13. An electroacoustic transducer plane array comprising:

a plurality of rectangular groups of transducers lying in a commonplane; the transducers in each of said groups being so arranged thateach of said transducers lies in ia horizontal row and a vertical columnof transducers; said rectangular groups being so arranged that theperpendicular bisectors of the sides of :the rectangles formed therebymeet at a common point; means for shading each of said groups so thatthe sensitivity of the transducers in the jp group is expressible as afunction of the number and sensitivities of the transducers in a linearray where said number of transducers in said line array equals twicethe number of said groups in said plane array and the sensitivities ofthe transducers in seid line array are shaded so as to increase from theextremities toward the center thereof according to the coefficients `ofra binomial probability distribution function, the sensitivity of thetransducers in said jp group being expressible as a function of thenurnber and sensitivities of the transducers in said line .arrayaccording to the equation tL SV sin S, nir z=jt|i Winx l) where jpidentifies any one of said groups by the number of groups said one ofsaid groups is removed from said common point and :1, 2, 3 tp, tpzthetotal number of groups, Sju=the sensitivity of the transducers in the jpgroup, jL identies any one of the transducers in said line array by thenumber of transducers said one of said transducers is removed from thecenter of said line array and =jp, njLzthe number of transducers the jLtransducer is removed from the center of said line array, SjL=thesensitivity of the jL transducer, tL=onehalf the total number oftransducers in sald line array, x identifies any one of the transducersin said line array, nx=the number of ltransducers the x transducer isremoved from the center of said linz array, and Sxzthe sensitivity ofthe x transducer. 14. The elcctroacoustic transducer plane array ofclaim 13 wherein adjacent transducers in said rows, said co1- umns, andsaid line array are `all the same distance apart. 15. Theelectroacoustic plane array of claim 13 wherein all the transducers insaid plane yarray are electrically coupled in parallel.

t6. An electroacoustic transducer plane array having square symmetry, aconstant pressure, plane wave nearield, and an operating frequencycomprising:

a plurality of rectangular groups of transducers lying in a commonplane; the transducers in each of said groups being so arranged thateach of said transducers lies in a horizontal row and a vertical columnof transducers; means for shading each of said groups so that thesensitivity of the transducers in the jp group is expressible as afunction of the number and sensitivities of the transducers in a linearray having the same operating frequency as said plane array and havingthe same distance between adjacent transducers as the distance betweengroups of adjacent transducers in said plane array, said number oftransducers in said line array being equal to twice the number of saidgroups and 'the sensitivities of the transducers in said line arraybeing shaded so as to increase from the extremities to the centerthereof `according to the coefficients of a binomial probabilitydistribution function having the general formuia l5 n=the number ofindependent binomial trials, and r=l, 2, 3 n; the sensitivity of thetransducers in said jp group being eXpressible as a function of thenumber and sensitivities of the transducers in said line array accordingto the equation where jp identities any one of said groups by the numberof groups said one of said lgroups is removed from said common point and:1, 2, 3 tp, tpzthe total number of groups, Sju=the sensitivity of thetransducers in the jp group, jL identifies any one of the transducers insaid line array by the number of transducers said one of saidtransducers is removed from the center of said line array and =jp,nj=the number of transducers the jL transducer is removed from thecenter of said line array, SjL=the sensitivity of the jL transducer,tL=oneha1f the total number of transducers in said line array, xidentifies 4any one of the transducers in said line array, nx=the numberof Itransducers the x transducer is removed from the `center of saidline array, and Sxzthe sensitivity of the x transducer; said operatingfrequency being such that where dzthe distance between adjacenttransducers in said line array, t=the wavelength of said operatingfrequency, and R=1/ 2 the distance between the points each side of thecenter of said line array around which a symmetry exists with respect tothe sensitivities of the transducers on the respective sides of saidcenter whose sensitivities are affected by said shading.

i7. The plane array of claim 16 wherein:

each of said transducers of said plane array has approximately the sameresonant frequency;

said operating frequency is at least one octave below said resonantfrequency;

the size of said transducers of said plane array is small in comparisonto said distance between adjacent transducers of said plane array; and

whereby said plane array is acousticaily transparent.

1S. An antenna array comprising:

a plurality of antennae;

means for shading the sensitivities of said antennae so that theirsensitivities increase from the extremities toward the center of saidarray according to the coefficients of a summed binomial probabilitydistribution function having the general formula t?, @yorin where ftzthenumber of independent binomial trials and r- -1, 2, 3 n; so that saidarray has a constant electric ield, plane wave near-field extending overthe aperture of said array.

19. The antenna array of claim 1S wherein said array a line array.

20. The antenna array of claim 18 wherein said array a plane array.

21. An antenna line array comprising:

a plurality of antennae;

means for shading said antennae so that their sensitivities increasefrom the extremities toward the center of said line according to thecoeicients of a summed binomial probability distribution function sothat said array has a constant electric field, plane wave near-fieldextending along said line.

22. An antenna line array having a constant electric field, plane wavenear-field and an operating frequency comprising:

a plurality of antennae equidistantly spaced forming said line array;

-means rfor shading said antennae so that their sensitivi- !tiesincrease from the extremities toward the center of said line arrayaccording to the coeiiicients of a binomial probability distributionfunction having the general formula (wwwrun-rn n=the number ofindependent binomial trials, and r=1, 2,3...n; said operating frequencybeing such that @man where 23. An antenna plane array comprising:

a plurality of rows of antennae lying in a common plane;

each of said rows containing a plurality of antennae;

the antennae in each of said rows being aligned with the antennae ineach of every other of said rows so that a plurality of columns ofantennae are formed;

means for shading each of said antennae so that its sensitivitysatisfies the equation S=S1S2 where S=the sensitivity of the antenna inthe plane array; S1=tl1e sensitivity of the antenna in a first linearray which occupies the sarne position with respect to the otherantennae in said first line array as said antenna of said plane arrayoccupies with respect to the other antennae of the row in which it isincluded, said rst line array containing the same number of antennae ascontained within said row being spaced identically to said antennae insaid row, the sensitivities of the antenna in said rst line array beingshaded so as to increase from the extremities toward the center thereofaccording to the coefficients of a binomial probability distributionfunction; and S2=the sensitivity of the antenna in a second line arraywhich occupies the same position with respect to the other antennae insaid second line array as said antenna of said plane array occupies withrespect to the other antennae of the column in which it is included,said second line array containing the same number of antennae ascontained within said column being spaced identically to the antennae insaid column, the sensitivities of the antennae in said second line arraybeing shaded so as t increase from the extremities toward the centerthereof according to the coefficients of a binomial probabilitydistribution function.

24. The plane array of claim 23 wherein:

said plane array has the same number of rows as columns; and

the antennae in said first and second line arrays are shaded so thattheir sensitivities increase from the extremities toward the centers ofthe respective line arrays according to the coefcients of the samebinomial probability distribution function.

25. An antenna plane array having approximate circular symmetry, aconstant pressure, -plane wave near-field, and an operating frequencycomprising:

a plurality of equidistantly spaced rows of antennae lying in a commonplane;

said rows being aligned so that each antenna in a row is aligned in acolumn with an antenna from each of every other row;

means for shading each of said antennae so that its sensitivitysatisfies the equation S=S1S2 where S=the sensitivity of the antenna inthe plane array; S1=the sensitivity of the antenna in a first line arraywhich occupies the same lposition with respect to the other antennae insaid first line array as said antenna of .said plane array occupies withrespect to the other antennae of the row in which it is included, saidfirst line array having the same operating frequency as said plane arrayand containing the same number of antennae as contained within said rowspaced the same distance apart as the antennae in said row, the.sensitivities of the antennae in said first line array being shaded soas to increase from the extremities toward the center thereof accordingto the coetiicients of a binomial probability distribution functiohaving the general formula n=the number of independent binomial trails,and r=l, 2, 3 n; and S2=the sensitivity of the antenna in a second linearray which occupies the same position with respect to the otherantennae in said second line array as said antenna of said plane arrayoccupies with respect to the other antennae of the column in which it isincluded, said second line array having the sarne operating frequency assaid plane array and containing the same number of antennae as containedwithin said column of said plane array spaced the same distance apart asthe antennae in said column, the sensitivities of the antennae in saidsecond line array being shaded so as to increase from the extremitiestoward the center thereof according to the coeicients of a binomialprobability distribution function having said general formula; saidoperating frequency being such that where d=the distance betweenadjacent antennae in one of said line arrays, 7\=the wavelength of saidoperating frequency, and R=the distance between the points each side ofcenter of said one of said line arrays around which a symmetry existswith respect to the sensitivities of the antennae on the respectivesides of said center whose sensitivities are affected by said shading.

26. An antenna plane array comprising:

a plurality of rectangular groups of antennae lying in a common plane;

the antennae in each of said groups being so arranged that each of saidantennae lies in a row and a column of antennae;

said groups being so arranged that the perpendicular bisectors of thesides of the rectangles formed thereby meet at a common point;

means for shading each of said groups so that the sen- .sitivity of theantennae in the jD group is expressible as a function of the number andsensitivities of the antennae in a line array where said number ofantennae in said line array equals twice the number of said groups in.said plane array and the sensitivities of the antennae in said linearray are shaded so as to increase from the extremities toward thecenter thereof according to the coeiiicients of a binomial probabilitydistribution function, the sensitivities of the antennae in said jpgroup being expressible as a function of the number and sensitivities ofthe antennae in said line array according to the equation where jpidentifies any one of said groups by the number of groups said one ofsaid groups is removed from said common point and =1, 2, 3 the totalnumber of groups, Sjp=the sensitivity of the antenna in the jp group, iLidentities any one of the antennae in said line array by the number ofantennae said one of said antennae is removed from the center of saidline array and =jp, nJ-L=the number of antennae the jL antenna isremoved from the center of said line array, SjL=the sensitivity of theiL antenna, tL=onehalf the total number of antennae in said line array,x identities any one of the antennae in said line array, nX--the numberof antennae the x antenna is removed from the center of said line array,and Sx=the sensitivity of the x antenna.

. tp, rp:

a common plane;

the antennae in each of said groups being so arranged that each of saidantennae lies in a row and a co1- umn of antennae;

means for shading each of said groups so that the sensitivity of theantennae in the jp group is expressible as a function of the number andsensitivities of the antennae in a line array having the same operatingfrequency as said plane array and having the same distance betweenadjacent antennae as the distance between adjacent groups of antennae insaid plane array, said number of antennae in said line array being equalto twice the number of said groups and the sensitivities of the antennaein said line array being shaded so as to increase from the extremitiesto the center thereof according to the coefcients of a binomialprobability distribution function having the general formula r=1, 2, 3n; the sensitivity of the antennae in 2O said jp group being expressibleas a function of the number and sensitivities of the antennae in saidline array according to the equation 1 tL 1 'SW- m1, SIL z=j+1 m01:- 1)Sx where jp identifies any one of said groups by the number of groupssaid one of said groups is removed from said common point and =1, 2, 3the total number of groups, Sjp=the sensitivity of the antennae in thejp groups, jL identies any one of the antennae in said line array by thenumber of antennae said one of said antennae is removed from the centerof said line array and :iw njL=the number of antennae the jL antenna isremoved from the center of said line array, SjL=the sensitivity of thejL antenna, tL=onehalf the total number of antennae in said line array,x identifies any one of the antennae in said line array, nx=the numberof antennae the x antenna is removed from the center of said line array,and Sx: the sensitivity of the x transducer;

. fp, tp:

said operating frequency being such that where d=the distance betweenadjacent antennae in said line array, t=the wavelength of said operatingfrequency, and R=the distance between the points each said of the centerof said line array around which a symmetry exists with respect to thesensitivities of the transducers on the respective sides of said centerwhose sensitivities are alected by said shading.

References Cited UNITED STATES PATENTS 1,643,323 9/1927 Stone 343-8442,407,329 9/1946 Turner 340-8 2,407,643 9/ 1946 Batchelder 340-16 X2,419,562 4/ 1947 Kandoian. 2,698,927 1/1955 Parr 340-15 OTHERREFERENCES Dolph, Proc. of the I.R.E., June 1946, pp. 335-339.

RICHARD A. FARLEY, Primary Examiner.

